A digital communication channel must have linear-phase characteristic such that the transmitted signal will keep its waveform unchanged after the signal transmission. However, since the actual communication channels usually possess nonlinear phase, it is necessary to connect a phase-linearizing circuit (phase-circuit) to such a nonlinear phase channel for linearizing the whole phase characteristic. In this paper, we propose a new iterative optimization scheme that utilizes the second-order-cone-programming (SOCP) for designing a high-accuracy phase-circuit. By approximating the highly nonlinear design constraint as a second-order-cone (SOC) constraint, a phase-circuit can be successfully designed through iterating the SOCP scheme. As compared to other existing linear-programming (LP) design and SOCP design, this new iterative SOCP-based minimax design can further reduce the peak errors of the frequency response and phase response of the designed phase-circuit. As a consequence, the new design method can further improve the phase-circuit design accuracy. We will utilize two design examples to verify the performance improvements over other existing design techniques.